Least Squares Estimation in a Single Index Model with Convex Lipschitz link

We consider estimation and inference in a single index regression model with an unknown convex link function. We propose a Lipschitz constrained least squares estimator (LLSE) for both the parametric and the nonparametric components given independent and identically distributed observations. We prove the consistency and find the rates of convergence of the LLSE when the errors are assumed to have only $q \ge 2$ moments and are allowed to depend on the covariates. In fact, we prove a general theorem which can be used to find the rates of convergence of LSEs in a variety of nonparametric/semiparametric regression problems under the same assumptions on the errors. Moreover when $q\ge 5$, we establish $n^{-1/2}$-rate of convergence and asymptotic normality of the estimator of the parametric component. Moreover the LLSE is proved to be semiparametrically efficient if the errors happen to be homoscedastic. Furthermore, we develop the R package \texttt{simest} to compute the proposed estimator.